Establishment homomorphism

In the mathematical branch of ring theory, the insertion homomorphism (also substitution or evaluation homomorphism ) describes the unambiguous continuation of a ring homomorphism between two commutative rings with one to a homomorphism of the polynomial ring belonging to the domain of definition in one or more variables.

definition

Let it be a homomorphism of commutative rings with one . Furthermore, denote the corresponding polynomial ring in a variable. ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle A[X]}$ ${\displaystyle A}$

A mapping can now be defined for each , which is a polynomial ${\displaystyle b\in B}$ ${\displaystyle \varphi _{b}\colon A[X]\to B}$

${\displaystyle f=\sum _{i\geq 0}a_{i}X^{i}}$

maps on

${\displaystyle \varphi _{b}(f):=\sum _{i\geq 0}\varphi (a_{i})b^{i}}$.

The homomorphism defined in this way is referred to as an insertion homomorphism.

properties

The following applies to all of them , i.e. it continues the homomorphism onto the polynomial ring if constant polynomials are identified with their coefficients derived from . Furthermore applies what the name Investiture motivated homomorphism: One is the concrete ring member for by symbolized variable one . ${\displaystyle \varphi _{b}(a)=\varphi (a)}$ ${\displaystyle a\in A}$ ${\displaystyle \varphi _{b}}$ ${\displaystyle \varphi }$ ${\displaystyle A[X]}$ ${\displaystyle A}$ ${\displaystyle \varphi _{b}(X)=b}$ ${\displaystyle b\in B}$ ${\displaystyle X\in A[X]}$

The fact that the homomorphism defined in this way always exists under the given conditions and is also uniquely determined is precisely what the proposition about the establishment homomorphism says .

Generalization to polynomial rings in several variables

Polynomial rings in finitely many variables

If there is a commutative ring with one, then polynomial rings can be inductively defined in a finite number of variables: starting from the polynomial ring , this initially arises by allowing polynomials with coefficients from . The further steps are carried out in the same way. ${\displaystyle A}$ ${\displaystyle A[X_{1}]}$ ${\displaystyle A[X_{1},X_{2}]:=A[X_{1}][X_{2}]}$ ${\displaystyle A[X_{1}]}$

Is now a homomorphism of commutative rings with one and to be associated in the polynomial variable, it can be to each tuple in an imaging define the polynomial ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle A[X_{1},\dots ,X_{n}]}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle (b_{1},\dots ,b_{n})}$ ${\displaystyle B}$ ${\displaystyle \varphi _{(b_{1},\dots ,b_{n})}\colon A[X_{1},\dots ,X_{n}]\to B}$

${\displaystyle f=\sum _{i_{1},i_{2},\dots ,i_{n}\geq 0}a_{i_{1}i_{2}\dots i_{n}}X_{1}^{i_{1}}X_{2}^{i_{2}}\dots X_{n}^{i_{n}}}$

maps on

${\displaystyle \varphi _{(b_{1},\dots ,b_{n})}(f):=\sum _{i\geq 0}\varphi (a_{i_{1}i_{2}\dots i_{n}})b_{1}^{i_{1}}b_{2}^{i_{2}}\dots b_{n}^{i_{n}}}$.

Polynomial rings in infinitely many variables

For a commutative ring with one , polynomials in infinitely many variables can be understood as maps ${\displaystyle A}$

${\displaystyle f\colon \mathbb {N} ^{I}\to A}$,

with an arbitrary index set is and the set of all functions of by a finite amount of support . The ring of polynomials is called in infinitely many variables with . ${\displaystyle I}$ ${\displaystyle \mathbb {N} ^{(I)}}$ ${\displaystyle I}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle A}$ ${\displaystyle A[(X_{i})_{i\in I}]}$

For a homomorphism between commutative rings with unity can be at any family in a mapping to define which is a polynomial maps to ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle \beta :=(b_{i})_{i\in I}}$ ${\displaystyle B}$ ${\displaystyle \varphi _{\beta }\colon A[(X_{i})_{i\in I}]\to B}$ ${\displaystyle f\in A[(X_{i})_{i\in I}]}$

${\displaystyle \varphi _{\beta }(f):=\sum _{\alpha }\varphi \left(f(\alpha )\right)b_{\alpha }}$,

where and . ${\displaystyle \alpha :=(a_{i})_{i\in I}\in \mathbb {N} ^{(I)}}$ ${\displaystyle b_{\alpha }:=\prod _{i\in I}b_{i}^{(a_{i})}}$

This case contains the cases for polynomials in one or finitely many variables. For this purpose, we consider a single-element or a finite index set . ${\displaystyle I}$

Point evaluation as a special case

If there is an injective ring homomorphism , i.e. if it is a ring expansion of , then in this special case the associated insertion homomorphism is also called point evaluation . In this case, one often writes for the value of at that point . ${\displaystyle \iota \colon A\to B}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle b\in B}$ ${\displaystyle \iota }$ ${\displaystyle \iota _{b}}$ ${\displaystyle f(b):=\iota _{b}(f)}$ ${\displaystyle f}$ ${\displaystyle b}$

The picture is often referred to as . The image is the smallest sub-ring of , which contains both the image and . It consists of all polynomial expressions of the form . ${\displaystyle \iota _{b}(A)}$ ${\displaystyle A[b]}$ ${\displaystyle B}$ ${\displaystyle \iota (A)\cong A}$ ${\displaystyle b}$ ${\displaystyle a_{0}+a_{1}b+\cdots +a_{n}b^{n}}$

There is a one , so true, so is referred to as a zero of . Of particular importance for the theory of algebraic equations is the core of the mapping for an element from which is not necessarily in . Is injective, so if and only if the zero polynomial is, it is also called transcendent over and it is isomorphic to . Otherwise one calls algebraic about , which is synonymous with the fact that the zero of a polynomial unequal to the zero polynomial with coefficients from occurs. ${\displaystyle f\in A[X]}$ ${\displaystyle a\in A}$ ${\displaystyle \iota _{a}(f)=0}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle \iota _{b}}$ ${\displaystyle b}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle \iota _{b}}$ ${\displaystyle \iota _{b}(f)=0}$ ${\displaystyle f}$ ${\displaystyle b}$ ${\displaystyle A}$ ${\displaystyle A[X]}$ ${\displaystyle A[b]}$ ${\displaystyle b}$ ${\displaystyle A}$ ${\displaystyle b}$ ${\displaystyle A}$

As in the case of the existence and uniqueness theorem, there are also direct generalizations to polynomial rings in several variables for point evaluation and all related terms.

Examples

If an ideal is in a ring (commutative and with one element), the homomorphism , which is composed of the projection onto the factor ring and the embedding in the associated polynomial ring , induces a ring homomorphism . The coefficients of a polynomial are thus reduced modulo . Here the monom is substituted by the corresponding monom from . ${\displaystyle I}$ ${\displaystyle A}$ ${\displaystyle \varphi \colon A\to A/I\hookrightarrow (A/I)[X]}$ ${\displaystyle A/I}$ ${\displaystyle (A/I)[X]}$ ${\displaystyle \varphi _{X}\colon A[X]\to (A/I)[X]}$ ${\displaystyle f\in A[X]}$ ${\displaystyle I}$ ${\displaystyle X\in A[X]}$ ${\displaystyle (1+I)X}$ ${\displaystyle (A/I)[X]}$

literature

• Siegfried Bosch : Algebra . 8th edition. Springer Verlag , Berlin 2013, ISBN 978-3-642-39566-6 , pp. 38 , doi : 10.1007 / 978-3-642-39567-3 .
• Jens Carsten Jantzen , Joachim Schwermer : Algebra . 2nd Edition. Springer Verlag , Berlin 2013, ISBN 978-3-642-40532-7 , pp. 104, 112-114 , doi : 10.1007 / 978-3-642-40533-4 .

Individual evidence

1. ↑ Jens Carsten Jantzen, Joachim Schwermer: Algebra. Springer-Verlag, 2013, ISBN 978-3-642-40533-4 , p. 113 ( limited preview in Google book search).
2. ^ Günter Scheja: Textbook of Algebra. Springer-Verlag, 2013, ISBN 978-3-322-80092-3 , p. 24 ( limited preview in Google book search).